midterm2_340

midterm2_340 - may use the identity 2 cos x sin x = sin 2 x...

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Common II, MATH 340 November 9, 2010 Prof. Boubendir 1. (20pts) Which of the following iterations will converge to the indicated α . If it does converge, determine the order of convergence. (a) x n +1 = 5 x 2 n - x n + 1 5 x n α = 1 (b) x n +1 = 15 x 2 n - 20 x n + 9 4 x n α = 1 (c) x n +1 = 4 ln( x n + 1) + 2 ( x n + 1) + 2 tan - 1 ( x n ) - 2( x n + 1) 2 α = 0 2. (20pts) Consider the function f ( x ) = e - ( x - 1) 2 . (a) Find the linear interpolating polynomial P 1 ( x ) at the points 0 and 1. (b) Find the quadratic interpolating polynomial P 2 ( x ) at the points 0, 0 . 5 and 1. (c) Bound the error f ( x ) - P 1 ( x ) on [0 , 1]. (d) Bound the error f ( x ) - P 2 ( x ) on [0 , 1]. 3. (20pts) Find a , b , c and d such that s ( x ) = ± ax 3 - 3 x on [0 , 1] bx 3 + cx 2 + d on [1 , 2] (1) is a natural cubic spline (a spline is natural if s 00 ( x ) = 0 at the endpoints). 4. (20pts) Use 4 subintervals ( n = 4) with Trapeziodal rule to approximate R 8 4 ( x - 4) 2 dx and bound the error of the approximation. Compare with the asymptotic error. 5. (20pts) Use 4 subintervals ( n = 4) with Simpson’s rule to approximate R 2 0 sin 2 ( πx ) dx and bound the error of the approximation. Compare with the asymptotic error. Note that you
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Unformatted text preview: may use the identity 2 cos x sin x = sin 2 x . 1 Formulas-h 2 12 ( b-a ) f 00 ( c )-h 2 12 ( f ( b )-f ( a ))-h 4 180 ( b-a ) f (4) ( c )-h 4 180 ( f 000 ( b )-f 000 ( a )) h [ 1 2 f ( x ) + f ( x 1 ) + f ( x 2 ) + ... + f ( x n-1 ) + 1 2 f ( x n )] h 3 [ f ( x ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + 2 f ( x 4 ) + ... + 2 f ( x n-2 ) + 4 f ( x n-1 ) + f ( x n )] ( x-x )( x-x 1 ) ... ( x-x n ) ( n + 1)! f ( n +1) ( c ) ( x-x )( x-x 1 ) ... ( x-x n ) ( n + 1)! f ( n +1) ( c ) L i ( x ) = ( x-x ) ... ( x-x i-1 )( x-x i +1 ) ... ( x-x n ) ( x i-x ) ... ( x i-x i-1 )( x i-x i +1 ) ... ( x i-x n ) P n ( x ) = y L ( x ) + y 1 L 1 ( x ) + ... + y n L n ( x ) 2...
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midterm2_340 - may use the identity 2 cos x sin x = sin 2 x...

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